Harnessing microcomb-based parallel chaos for random number generation and optical decision making

Optical chaos is vital for various applications such as private communication, encryption, anti-interference sensing, and reinforcement learning. Chaotic microcombs have emerged as promising sources for generating massive optical chaos. However, their inter-channel correlation behavior remains elusive, limiting their potential for on-chip parallel chaotic systems with high throughput. In this study, we present massively parallel chaos based on chaotic microcombs and high-nonlinearity AlGaAsOI platforms. We demonstrate the feasibility of generating parallel chaotic signals with inter-channel correlation <0.04 and a high random number generation rate of 3.84 Tbps. We further show the application of our approach by demonstrating a 15-channel integrated random bit generator with a 20 Gbps channel rate using silicon photonic chips. Additionally, we achieved a scalable decision-making accelerator for up to 256-armed bandit problems. Our work opens new possibilities for chaos-based information processing systems using integrated photonics, and potentially can revolutionize the current architecture of communication, sensing and computations.


calculated by
Where δI(t) = I(t) − ⟨I(t)⟩ t . For ease of presentation, the full width at half maximum (FWHM) of the ACF is employed to qualify the recorded data, as shown in Supplementary   Fig. 1a. A lower FWHM indicates a faster chaotic process, benefiting applications such as random bits generation and ranging.
Despite the ACF, the radio-frequency (RF) spectrum is another common characterization for the chaotic signal. As shown in Supplementary Fig. 1b, the chaotic signal obtained by detecting a comb line of the chaotic combs shows a continuous spectrum in the RF domain, with a decreasing power density apart from the zero frequency. Thus, a 10 dB-bandwidth is employed here to qualify the chaotic signal in the RF domain.
Employing the FWHM of the ACF and the 10 dB-bandwidth of the RF spectrum, we could qualify a single chaotic signal. The bandwidth is one of the most important characteristics of chaotic signal. For the detection of chaotic signals, the sampling rate should be larger than the twice of the signal bandwidth. While for applications such as random bit generation, the sampling rate can be larger than the twice of the signal bandwidth, with post-processing to obscure inter-sampling correlation [1]. While this will not increase the rate of entropy production, which limits the random bit generation rate. The entropy rate can be estimated by: Where τ is the sampling period, BW is the bandwidth of the entropy source, N ϵ is the number of bits per sample, p(x) is the probability density function of the entropy source, u(x) is the uniform distribution and D KL is the Kullback-Leibler divergence from u(x) to p(x). We can see that the entropy is limited by the twice of bandwidth and the probability density function. The entropy can be increased with a larger bandwidth and a more flat probability density function.
In this work, we are aimed at the realization of a massively parallel chaotic signal generator. The orthogonality between channels is of great interest to prove the parallelism.
Considering two signals detected I m (t) and I n (t), the orthogonality is qualified by the crosscorrelation XCF m,n (τ ) = ⟨δI m (t + τ ) · δI n (t)⟩ t ⟨δI 2 m (t)⟩ t · ⟨δI 2 n (t)⟩ t Where δI m (t) = I m (t) − ⟨I m (t)⟩ t and δI n (t) = I n (t) − ⟨I n (t)⟩ t . If I m and I n are correlated at a delayed time τ , there will be a peak at XCF m,n (τ ) where the height values the strength of the correlation. Thus, the maximum of XCF m,n (τ ) could be employed to value the correlation or orthogonality between two detected signals. For the ease of the expression, the maximum of XCF m,n (τ ) is symbolized by XCF m,n . A lower XCF m,n indicates a lower correlation or a better orthogonality between I m (t) and I n (t). In the experiment, the comb lines are recorded individually by an oscilloscope to obtain I m (t) and a electrical spectrum analyzer to obtainĨ m (f )). In the simulation, the comb evolution is simulated by the Lugiato-  Supplementary note III Experiment: Route into the chaotic state Supplementary Fig. 4a shows the 90 GHz FSR AlGaAs ring resonator tested, working in anomalous dispersion. By sweeping a continuous-wave laser through a resonance, multiple comb lines can be stimulated with different comb states, termed as the primary comb state, the subcomb state, and the chaotic comb state, as reported previously. Also, the chaotic comb state can be estimated by the destabilization of the soliton state, as discussed in [4].
To observe the path from the coherent comb state to the chaotic comb state, the evolution Shown in Fig. 3h and Supplementary Fig. 5a, the inter-channel correlation is obtained in experiment. It is worth noting that there is an obvious correlation between symmetrical comb lines while the correlation between other comb lines is weak (less than 0.04) enough to be neglected. These properties are also captured in simulation as shown in Supplementary   Fig. 5b, where a pump power of around 100 mW is employed. In this section, we will show the influence of different parameters on the chaotic comb.
For simplification, different chaotic combs are characterized by the minimum full widths at half maximum (FWHM) of the autocorrelation of comb lines around the pump mode.
Supplementary Fig. 6 shows the results under different conditions. It is worth noting that the FWHM of autocorrelation function would be significantly affected by the quality factor and the pump power, while no obvious variation could be found as the variation of the second order dispersion. By increasing the pump power, the intracavity energy could be rapidly raised, resulting in a more chaotic state. Thus, it is reasonable that the second order dispersion will not influence the FWHM of the ACF. While the number of comb lines will be affected by the second order dispersion as the circumstance of soliton microcombs.
Different from the soliton microcomb, as shown in Supplementary Fig. 6b1, it seems that a high-quality factor is not a positive condition for chaotic combs. This can be explained by the effective coupling state as illustrated in [5]. where the coupling quality factors are varied from 10 5 to 10 8 with the intrinsic quality factor set to 10 7 . Under an over-coupling state (Q c < 10 7 ), a lower FWHM of ACF is ensured compared with the critical coupling state, indicating a better chaotic state. A higher conversion efficiency is also attractive as a higher comb line power is promised. Compared with the critical coupling state, the over-coupling state can deliver comb lines with 20 dB higher power. Supplementary Fig. 7e and f show the radio frequency spectra and autocorrelation functions of different comb lines (with mode number -1, -10, -50, -100 as shown in Supplementary Fig. 7d). It is worth noting that all comb lines feature an effective radio-frequency bandwidth above 10 GHz and an FWHM below 0.1 ns. Considering a sampling rate of 10 GSa/s, the correlation between adjacent sampling data is around 0.1, which is acceptable for a random bit generator. With a proper design, the total output rate of the chaotic-comb-based parallel random bit generator could reach 3 Tbps (considering 30 Gbps pre-channel).

Supplementary note VII Experiment and Simulation: Influence of the intermode coupling
Supplementary Fig. 8: Influence of inter-mode coupling. a, the optical spectrum of chaotic comb in the experiment shown in the main text. b, the integrated dispersion profile of the tested microring. c, the zoom-in optical spectrum at C band. d, the simulated optical spectrum considering the mode-crossing. e, the correlation between different comb lines in simulation. The auto-correlation function (f ) and radio frequency spectrum (g) with (blue) and without (red) mode crossing.
In Supplementary Fig. 7d, we can see flat optical spectra of chaotic combs, while the optical spectra observed in the experiment were fluctuating at the central part. Supplementary   Fig. 8c shows the zoom-in spectrum, where the power variation is approximately 7 dB.
The power variation could be induced by inter-mode coupling as the circumstance in the bright soliton. Supplementary Fig. 8b shows the measured integrated dispersion profile of the tested microring, where multiple anti-mode-corssing points can be observed. In the simulation, more than one transverse mode can be supported in the waveguide used here (400 nm × 650 nm). Coupling between different modes is possible as orthogonality between modes is not promised, causing shifted resonance frequency and varied quality factors. The influence of inter-mode coupling has been well studied in [6,7]. Summarily, the inter-mode coupling can induce power variation at comb lines of the chaotic comb. Supplementary Fig.   8d shows a simulated optical spectrum where an anti-mode-crossing is induced around mode 23. Supplementary Fig. 8e shows the correlation between different comb lines considering the inter-mode coupling. Despite localized dispersion variation, the great orthogonality between comb lines is not lost, agreeing with our experiment result. The autocorrelation function and radio-frequency spectrum at the crossing point are given in Supplementary Fig.   8f and Supplementary Fig. 8g. In this simulation case, it is worth noting that the presence of mode crossing results in a wider RF spectrum. This wider spectrum may be attributed to the broader resonance and lower quality of the microcomb caused by the mode crossing.
The impact of mode crossing on microcombs is a complex problem, which requires further investigation in future work.  Supplementary Fig. 9 shows the influence of the 3PA and the FCA on the chaotic properties. To take the 3PA into account, a modified Lugiato-Lefever equation with the equation describing the dynamics of the free carrier is employed: where the 3PA and free carrier absorption are considered as the second part at the right side of the Eq. S4. β 3P A is the 3PA coefficient, A ef f is the effective area of the waveguide, σ and µ are the FCA parameters, N c is the density of free carriers and τ ef f represents the effective life of free carriers. Considering an effective carrier lifetime of 5 ns, the chaotic signal will be degraded. As for the cross-correlation between symmetric comb lines, the correlation raised compared with the chaotic comb without the 3PA. More seriously, the autocorrelation property is also degraded with a larger FWHM and higher sidelobes. All this degradation can be attributed to a lower intracavity power as shown in Supplementary Fig.   9a. The simulation result can explain the difference between our experiment result where the FWHM under 130 mW is higher than that in simulation under the same condition. This indicates the chaotic property of chaotic combs in AlGaAsOI microrings is limited by the 3PA. This could be eased by inducing the PIN junction structure to shorten the effective carrier lifetime, which has been well studied in nonlinear optics based on Silicon-on-Insulator platform [8]. Considering an effective carrier lifetime of 0.1 ns, the chaotic signal is of the same quality as that without 3PA. The AlGaAsOI microring with PIN junction structures might be a more excellent platform for the chaotic comb generation, to deliver massively parallel chaotic signals with tens GHz chaotic bandwidth.

Supplementary note IX Experiment: Details of the test link
Supplementary Fig. 10: The test link of chaotic combs Supplementary Fig. 10 shows the general test link for the experiment. To stimulate the chaotic comb state, a high quality AlGaAsOI microring is pumped. To get a high pump power, an EDFA is employed to boost the pump laser to a high level and a bandpass filter is used to suppress the ASE noise. Then the amplified pump laser is injected into the microring after passing an isolator and a polarization controller. As illustrated above, by sweeping the frequency of the pump laser, we could obtain a chaotic comb as a massively parallel chaotic source. At the output side of the microring chip, a notch filter is used to suppress the pump mode and the remaining comb lines are boosted by a DWDM EDFA. Then the amplified comb lines are divided by the wavelength selective switch and sent to different photodetectors . Due to the insertion loss induced by the wavelength selective switch (6 dB), an EDFA is used before injecting into the PD. The detected signal is sent into an oscilloscope to observe the time domain variation and an electrical spectrum analyzer to obtain the RF domain spectrum. For the characterization of the cross-correlation, two comb lines are filtered and sent to two PDs respectively at the same time. Cross-correlation is employed to the two detected signals to value the correlation between two selected channels.
In the application for random bit generation, a silicon photonic wavelength division demultiplex receiver is employed. Fig. 3 shows the test link and Supplementary Fig. 11a shows the microscope image of the SiPh chip. The DWDM receiver is constituted of one 16-channel arrayed waveguide grating and 16 Ge-on-Si photodetectors. The optical loss from the input edge coupler to the photodetector is estimated to be 8∼9 dB, by testing a similar circuit where photodetectors are replaced by an array of edge couplers. As the amplified comb line is injected into the chip, the AWG will selectively divide the input signal into PD.
The tested transmission spectra of the 16-channel AWG are given in Supplementary Fig.   11b. The channel spacing is verified to be around 180 GHz, fit well with the channel spacing of the microcombs. In the experiment, the chaotic signal is detected by commercial InP photodetectors and Si-Ge photodetectors respectively. In this part, we show that the frequency response or bandwidth of the Ge-Si photodiode is worse than that of the commercial photodiode, which degenerates the recorded signal. Supplementary Fig. 12a shows the measured frequency response of the integrated Ge-Si photodiode and the InP photodiode. The Ge-Si photodiode shows a faster decrease at the high frequency. This is revealed at the frequency spectra of recorded signal as shown in Supplementary Fig. 12b. Supplementary Fig. 12c shows successfully. In addition, it is worth noting that the input power of -9 dBm is enough for the Ge-Si photodetector as illustrated in [9]. Considering a high conversion efficiency shown in Supplementary Fig. 6d where the power of each comb line could reach -5 dBm, a fully integrated chaotic signal detector without an inter-chip amplifier is possible. 14c shows the spectra of amplified combs. The first order amplification of the EDFA is set to the maximum. Different from the EDFA, the gain bandwidth of the SOA is wider. To obtain higher power in the C-band, a bandpass filter was employed after the first SOA to suppress comb lines outside the C-band. The filtered spectrum is shown in Supplementary   Fig. 14d. Due to a higher noise figure compared to the EDFA, the comb lines amplified by the SOA show lower SNR. For the comb line around 1545 nm, the SNR is 23.7 dB, which is 4.3 dB lower than that amplified by the EDFA. Moreover, due to the insertion loss of the bandpass filter (5 dB), the power amplified by the SOA is lower. Thus, the filtered comb is amplified by another SOA. Supplementary Fig. 14e shows the optical spectrum after the second SOA. In addition, for the EDFA, the second-order amplifier is turned on and set to the maximum. The output powers of the SOA and the EDFA are 16 dBm and 22 dBm, respectively. Thus, the comb line amplified by the SOA is weaker. One comb line around 1545 nm is filtered out by a tunable bandpass filter with 6 dB insertion loss. The comb line power sent to the photodiode is -3 dBm and -8 dBm for the EDFA and SOA, respectively.
Supplementary Fig. 14f shows the radio frequency spectra amplified by the EDFA and SOA.
Due to a higher noise figure and lower saturation power, the signal amplified by the SOA shows a 3.1 dB lower SNR. Supplementary Fig. 14g shows As we have discussed in Methods, the selection of k value is vital for decision makers. Thus, we sweep the k value for different problem scales, and the convergence cycles are recorded as shown in Supplementary Fig. 15a. To show the influence of k on the convergence speed, the evolution of the corrected decision rate with the increase of cycles under different k values are shown in Supplementary Fig. 15b, where N=16. It is obvious that a higher convergence speed is accompanied with a larger k at the beginning of the decision process.
While the corrected decision rate could not reach 100% as the game carrying on. It is caused by worry decisions as shown in Supplementary Fig. 15c, where one decision process with k=0.4 is presented. Due to a large k, the decision maker only tests 5 slot machines and a high bias value is added to the 25th channel, making a premature decision without sufficient exploration. In addition, one decision process with k=0.1 is shown in Supplementary Fig.   15d. Under a small k, the decision maker will be more cautious, causing a slow convergence